Math 100 Homework 1
fall 2010
due 15/9
1. (11.1, 24) Describe the surface whose equation is given by
x2 + y2 + z 2 y = 0.
2. (11.3, 22) Find the acute angle formed by two diagonals of a cube.
3. (11.3, 28) Determine if it is true that for any vectors a, b
Math 100 Homework 7
fall 2010
1. (a) C can be parametrized by r(t) = (1 t)a + tc, (1 t)b + td) for
0 t 1. Thus,
ydx + xdy
1
= 0 [(1 t)b + td](c a) + [(1 t)a + tc](d b)]dt
= ad bc.
C
(b) Let C1 , C2, C3 be the line segments joining (x1 , y1) to (x2 , y2),
Math100, 3-Dimensional Space; Vectors
1.1
Rectangular Coordinates
Points in the space are presented by their coordinates, similar to the case for
points in a plane. Take three lines in the space which are mutually perpendicular
and we call them the x-axis
Math100, Vector-Valued Functions
2.1
Curves
Denition 2.1 A parametric representation of a curve (or a curve) is a function r, in which a number is plugged in and a vector is returned (or a vectorvalued function).
Remark 2.2 Let r be a vector-valued functi
Math100, Partial Derivatives
3.1
Functions of Two or More Variables
Denition 3.1 f is a function in n variables if f is a function in which a
vector in the n-dimensional space is plugged in and a number is returned.
Denition 3.2 Let f be a function in n v
Math100, Multiple Integrals
4.1
Double Integrals
Denition 4.1 Let f be a function in two variables, R be a region in the plane.
The integral of f over R, denoted by R f or R f (x, y)dxdy, is dened to be the
limit of
i
f (a point in the ith rectangle) (are
Math100, Vector Calculus
5.1
Vector Fields
Denition 5.1 A vector eld in the n dimensional space is a function in which
a vector in the n dimensional space is plugged in and a vector in the n dimensional space is returned.
Example 5.2 Let be a function in
Math100, Vector Calculus
5.1
Vector Fields
Denition 5.1 A vector eld in the n dimensional space is a function in which
a vector in the n dimensional space is plugged in and a vector in the n dimensional space is returned.
Example 5.2 Let be a function in
Math2011, 3-Dimensional Space; Vectors
1.1
Rectangular Coordinates
Points in the space are presented by their coordinates, similar to the case for
points in a plane. Take three lines in the space which are mutually perpendicular
and we call them the x-axi
Math2011, Vector-Valued Functions
2.1
Curves
Denition 2.1 A parametric representation of a curve (or a curve) is a function r, in which a number is plugged in and a vector is returned (or a vectorvalued function).
Remark 2.2 Let r be a vector-valued funct
Math2011, Partial Derivatives
3.1
Functions of Two or More Variables
Denition 3.1 f is a function in n variables if f is a function in which a
vector in the n-dimensional space is plugged in and a number is returned.
Denition 3.2 Let f be a function in n
Math2011, Multiple Integrals
4.1
Double Integrals
Denition 4.1 Let f be a function in two variables, R be a region in the plane.
The integral of f over R, denoted by R f or R f (x, y)dxdy, is dened to be the
limit of
i
f (a point in the ith rectangle) (ar
Math 100 Homework 7
fall 2010
due 6/12
1. (15.4, 38)
(a) Let C be the line segment from a point (a, b) to a point (c, d). Show
that
ydx + xdy = ad bc.
C
(b) Use the result in part (a) to show that the area A of a triangle with
successive vertices (x1 , y1
Math 100 Homework 6
fall 2010
1. The solid G collects points (x, y, z ) satisfying
0zy
0 y 4 x2
2 x 2.
and
and
Therefore,
G
=
ydV
4x2
y =0
4x2
y =0
1
3 (4
2
x=2
2
x=2
2
x=2
=
=
= 4096/105.
y
z =0
2
ydzdydx
y dydx
x2 )3 dx
2. The solid G collects points (
Math 100 Homework 1
fall 2010
due 15/9
1. (11.1, 24) Describe the surface whose equation is given by
x2 + y2 + z 2 y = 0.
2. (11.3, 22) Find the acute angle formed by two diagonals of a cube.
3. (11.3, 28) Determine if it is true that for any vectors a, b
Math 100 Homework 1
fall 2010
1. If (x, y, z ) belongs to the given surface,
x2 + y 2 + z 2 y = 0
x2 + y 2 y + 1 + z 2 = 1
4
4
1
x2 + (y 1 )2 + z 2 = 4
2
1
1
the distance from (x, y, z ) to (0, 2 ) is 2 .
Therefore, the given surface is the sphere centere
Math 100 Homework 1
fall 2010
due 30/9
1. (11.6, 18a) Determine whether the line
x = 3t; y = 7t; z = t
intersects the plane
2x y + z + 1 = 0.
If so, nd the coordinates of the intersection.
2. (11.6, 30) Find an equation of the plane through the points P1(
Math 100 Homework 2
fall 2010
1. Let P be a point on the given line, then there is a number t such that
P = (3t, 7t, t). If P is on the given plane also, we have 2(3t) 7t + t +1 = 0
or 1 = 0! which is impossible. Therefore the given line and the given
pla
Math 100 Homework 3
fall 2010
due 15/10
1. (12.3, 8) Find the arc length of the curve
11
1
r(t) = ( t, (1 t)3/2, (1 + t)3/2 )
23
3
1 t 1.
2. (12.3, 30) Find an arc length parametrization of the curve
r(t) = (sin et , cos et, 3et ) t 0.
3. (13.1, 56) Desc
Math 100 Homework 3
fall 2010
1. The arc length of the given curve is
1
|r (t)|dt
1
1
11
1
= 1 |( 2 , 2 1 t, 2 1 + t)|dt
11
= 2 1 12 + 1 t + 1 + tdt
1
= 23 1 dt
=
2. Let
3.
t
g(t) = 0 |r (s)|ds
t
= 0 |(es cos es , es sin es , 3es )|ds
t
= 0 2es ds
= 2(et
Math 100 Homework 4
fall 2010
due 29/10
1. (13.6, 74) On a certain mountain, the elevation z above a point (x, y) in
an xy-plane at sea level is
z = 2000 0.02x2 0.04y2,
where x, y and z are in meters. The positive x-axis points east, and the
positive y-ax
Math 100 Homework 4
fall 2010
1. Let f (x, y) = 2000 0.02x2 0.04y2, p = (20, 5, 1991)
(a) The directional derivative of f along (1, 0) (the unit vector pointing
to the east) at p is
D(1,0)f (p) =
f
(20, 5) = 0.04(20) > 0.
x
The climber would ascend by goi
Math 100 Homework 5
fall 2010
due 12/11
1. (14.2, 42) Evaluate the volume of the solid enclosed by y 2 = x, z = 0, and
x + z = 1.
2. (14.2, 50) Express
ln x
e
f (x, y)dydx
0
1
as an equivalent integral with the order of integration reversed.
3. (14.3, 28)
Math 100 Homework 5
fall 2010
1. Let f be a function in two variables dened by f (x, y) = 1 x, R is the
region consisting of points (x, y) satisfying y 2 x 1 and 1 y 1.
Then, f (x, y) 0 when (x, y) belongs to R, and the given solid is the
collection of po
Math 100 Homework 6
fall 2010
due 26/11
1. (14.5, 10) Let G be the solid enclosed by the plane z = y, the xy-plane,
and the parabolic cylinder y = 4 x2 . Evaluate the triple integral
ydV
G
2. (14.5, 20b) Let G be the solid enclosed by the surfaces dened b
Math2011, Vector Calculus
5.1
Vector Fields
Denition 5.1 A vector eld in the n dimensional space is a function in which
a vector in the n dimensional space is plugged in and a vector in the n dimensional space is returned.
Example 5.2 Let be a function in